Realizing the Teichm\"uller space as a symplectic quotient

TL;DR

This paper constructs an infinite-dimensional symplectic structure on the space of Riemannian metrics compatible with a volume form on a closed surface, and shows that Teichmüller space can be realized as a symplectic quotient.

Contribution

It introduces a novel symplectic framework for Teichmüller space using group-valued momentum maps and symplectic reduction techniques.

Findings

Teichmüller space is a symplectic orbit reduced space.

The group of volume-preserving diffeomorphisms acts with a natural momentum map.

The moduli space of Riemann surfaces can be realized via symplectic reduction.

Abstract

Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving diffeomorphisms by push-forward has a group-valued momentum map that assigns to a Riemannian metric the canonical bundle. We then deduce that the Teichm\"uller space and the moduli space of Riemann surfaces can be realized as symplectic orbit reduced spaces.